EXOTIC IN TOPOLOGICAL SOLITON MODELS. V.B.Kopeliovich Institute for Nuclear Research of RAS, Moscow, 117312

The status and properties of discovered recently are discussed within topological soliton and simplistic model.

1 Discovery of pentaquarks

One of striking events in elementary physics after latest conference was discovery of resonances with unusual properties (end of 2002 - 2004): + Θ (1540), strangeness S = +1, isospin I = 0 (most likely), width ΓΘ < 10 Mev, seen by different collaborations in Japan, Russia, USA, FRG, CERN; −− Ξ3/2 (1862), S = 2 , I = 3/2 (?), Γ < 18 Mev observed by NA49 Collab. at CERN; 0 − P Θc(3099), charm C = 1, Γ < 15 Mev seen by H1 Collab., DESY. Spin-parity J of these states is not measured− yet. These states are manifestly exotic because they cannot be made of 3 valence quarks only. There are different possibilities to have states: a) positive strangeness S > 0 (or negative charm C < 0, or positive beauty), since s-quark has S = 1 and c-quark C = +1, b) large (in modulus)− negative strangeness S < 3B, B - baryon number; similar for charm or beauty, − c) large enough isospin I > (3B + S)/2 , if 3B S 0, or charge Q > 2B + S or − ≤ ≤ Q < B in view of Gell-Mann - Nishijima relation Q = I3 + Y/2. − + The pentaquarks Θ (1540) and Θc(3099), if it is confirmed, are just of the type a), the possibility b) is difficult to be realized in practice, since large negative strangeness of produced baryon should be balanced by corresponding amount of pos- −− itively strange . The state Ξ3/2 is of the type c). The minimal quark contents + −− 0 of these states are: Θ = (dduus¯); Ξ = (ssddu¯); Θc = (dduuc¯), and by this reason they are called pentaquarks. Chronology of pentaquarks discovery.

Θ+, S = +1 baryon was observed first at SPring-8 installation (RCNP, Japan)[1] in reaction γ12C K+n+.... The reported mass is 1540 10 Mev and width Γ < 25 Mev, confidence level→(CL) 4.6σ. Soon after this and independently DIANA collaboration at ITEP, Moscow [2] observed Θ+ in interactions of K+ in Xe bubble chamber. The mass of the bump in K0p invariant mass distribution is 1539 2 Mev, Γ < 9 Mev, confidence level a bit lower, about 4.4σ. Confirmation of thisresult came also from several other experiments [3, 4, 5, 6, 7, 8, 9] mostly in reactions of photo-(electro)- production. The CLAS Collaboration [10] provided recently evidence for two states in Θ-region of the K+n invariant mass distribution at 1523 5 Mev and 1573 5 Mev. The nonobservation of Θ+ in old - scattering data provided restric- tion on the width of this state. Phase shift analysis of KN-scattering in the energy interval 1520 1560 Mev gave a restriction Γ < 1 Mev [11]. Later analysis of total cross sections −of kaon-deuteron interactions allowed to get the estimate for the width of Θ+: Γ 0.9 0.2 Mev [12]. '  The doubly strange cascade Ξ3/2, S = 2, probably with isospin I = 3/2 is observed in one experiment at CERN, only, −in -proton collisions at 17Gev[13]. The mass of resonance in Ξ−π− and Ξ¯+π+ systems is 1862 2 Mev and mass of resonance in Ξ−π+, Ξ¯+π− systems is 1864 5 Mev, width Γ <18 Mev, and CL = 4.0σ. The fact makes this result more reliablethat resonance Ξ−− was observed in antibaryon channel as well. However, this resonance is not confirmed by HERA-B, ZEUS, CDF, WA-89 collaborations (see e.g. [14]), although there is no direct contra- diction with NA49 experiment: upper bounds on the production cross sections of Ξ3/2 are obtained in this way. 0 The anticharmed Θc, C = 1 was observed at H1, HERA, Ger- many, in ep collisions. The mass of resonance in− D∗−p, D∗+p¯ systems is 3099 3 5 Mev, Γ < 15 Mev, CL = 6.2σ. ZEUS collaboration at HERA [16] did notcon- 0 firm the existence of Θc with this value of mass, and this seems to be already serious contradiction, see again [14]. There is also some evidence for existence of other baryon states, probably exotic, 0 e.g. nonstrange state decaying into nucleon and two [17] or resonance in ΛKS system with mass 1734 6 Mev, Γ < 6 Mev at CL = (3 6)σ which is N ∗0 or Ξ∗ ,  − 1/2 observed by STAR collaboration at RHIC [18] in reaction Au+Au at √sNN 200 Gev. Already after the Quarks-04 conference the negative results on Θ+ observ∼ ation have been reported by several groups. If high statistics experiments will not confirm existence of Θ+, it would be interesting then to understand why about 10 different ex- periments, although each of them with not high statistics, using different installations and incident , provided similar positive results. Information about status of higher statistics experiments performed or to be performed at JLAB (CLAS Collab.) can be found in [19].

2 Early predictions

From theoretical point of view the existence of such exotic states by itself was not unexpected. Such states have been discussed first within MIT quark-bag model [20]. The mass of these states was found considerably higher than that reported now: MΘ P − ' 1700 Mev, J = 1/2 . These studies were continued by other authors [21]: MΘ 1900 Mev, J P = 1/2− and [22]. From analysis of existed that time data on KN' interactions the estimate was obtained in [23] MΘ 1705 Mev (I = 0), 1780 Mev (I = 1) with very large width. ' In the context of chiral soliton model the 10¯ and 27 -plets of exotic baryons were mentioned first in [24], without any mass{estimates,} { ho}wever. Rough estimate within the ”toy” model, M10¯ M8 600 Mev, was made a year later in [25]. A resonance-like behaviour of KN− scattering' phase shift in Θ channel was obtained in [26] in a version of Skyrme model (in the limit MK = Mπ). Numerical estimate MΘ 1530 Mev was obtained by M.Praszalowicz [27], but it was no serious grounds for this' since mass splittings within octet and decuplet of baryons was not described here satisfactorily (”accidental” prediction). Extension of quantization condition [28] to ”exotic” case was made in [29] where masses of exotic baryonic systems (B arbitrary, Nc = 3) were estimated as function of the number of additional quark-antiquark pairs, or ”exoticness number m”: ∆E 2 ∼ m/ΘK, m /ΘK. But here it was no mass splittings estimates inside of multiplets, no discussion of masses of particular baryons. First calculation with configuration mixing due to flavor symmetry breaking (mK = mπ) was made in [30] where mass splittings within the octet and decuplet of baryons6 were well described, and estimate obtained M 1660 Mev. ”Strange” or Θ ' kaonic inertia ΘK was underestimated in this work, as it is clear now. The estimate M 1530 Mev, coinciding with [27], and first estimate of the Θ ' width, ΓΘ < 30 Mev were made 5 years later in [31]. It was ”a luck”, as stated much later by same authors (hep-ph/0404212): mass splitting inside of 10¯ was obtained greater than for decuplet of baryons, 540 Mev, and it was supposed that resonance N ∗(1710) 10¯ , i.e. it is the nonstrange component of antidecuplet. The above mass value was∈a {result} of subtraction, 1530 = 1710 540/3. [31] is instructive example − of a paper which, strictly speaking, is not quite correct, but being in right direction, stimulated successful searches for Θ+ in RCNP (Japan) and ITEP (Russia). Skyrme-type model with vibrational modes included was studied first in [32] with a result MΘ 1570 Mev, ΓΘ 70 Mev. A mistake (or misprint?) in paper’s [31] width estimate'was noted here. ∼

Developements after Θ+ discovery.

After discovery of pentaquarks there appeared big amount of papers on this subect which develop theoretical ideas in different directions: within chiral soliton models [33, 34] and many other, phenomenological correlated quark models [35, 36, 37] and other, QCD sum rules [38], by means of lattice calculations [39], etc. It is not possible to describe all of them within restricted framework of this talk (reviews of that topic from different sides can be found, e.g. in [42, 46]). Quite sound criticism concerning chiral soliton models was developed in [40, 41], but it should be kept in mind that the drawbacks of soliton approach should be compared with uncertainces and drawbacks of other models. There is no regular way of solving relativistic many-body problem of finding bound states or resonances in 3-,5-, etc. quark system, and the chiral soliton approach, in spite of its drawbacks, provides a way to circumwent some of difficulties. The correlated quark models, -triquark model [35], or diquark- diquark-antiquark model [36], being interesting and predictive, contain good amount of phenomenological guess.

3 Topological soliton model

In spite of some uncertainties and discrepances between different papers, the chiral soliton approach provided predictions for the masses of exotic states near the value observed later, considerably more near than quark or quark-bag models. Here I will be restricted with this model, mainly. Situation is somewhat paradoxical: it is easier to estimate masses of exotic states within chiral soliton models, whereas interpretation is more convenient in terms of simplistic . In topological (chiral) soliton models the baryons and baryonic systems appear as classical configurations of chiral (”pionic” in simplest SU(2) version) fields which are characterized by the topological or winding number identified with the baryon number of the system [43]. This baryon number is the 4-th component of the Noether current generated by the Wess-Zumino term in the action written in a compact form by Witten [44], I shall not reproduce it here. In other words, the B-number is degree of the map R3 SU(2), or R3 S3, since SU(2) is homeomorphic to 3-dimensional sphere S3: → → 1 2 (f, α, β) 3 B = − 2 sf sαI d r (1) 2π Z  (x, y, z)  where functions f, α, β, describing SU(2) , define the direction of unit vector ~n on 3-dimensional sphere S3 and I[(f, α, β)/(x, y, z)] is Jacobian of corresponding transformation. More details can be found, e.g. in [44, 33, 46]. It is important that the number of dimensions of the ordinary space, equal to 3, coincides with the number of degrees of freedom (or generators) of the SU(2) group, and this makes possible the mapping ordinary space onto isospace. This can be explanation why the isospin symmetry takes place in hadronic world. For each value of baryon number one should find the classical field configuration of minimal energy (mass) - this is done often by means of variational minimization numerical codes. For B = 1 configuration of minimal energy is of so called ”hedgehog” type, where chiral field at each space point can be directed along radius vector drawn from center of soliton, for B = 2 it has torus-like form, for B = 3 it has topology of tetrahedron, etc. The next step is quantization of these configurations to get spectrum of states with definite quantum numbers, isospin I, strangeness S or hyperchrge Y . In the collective coordinates quantization procedure [48, 28] one introduces the angular velocities of rotation of skyrmion in the SU(3) configuration space, ωk, k = 1, ...8: † A (t)A˙(t) = iωkλk/2, λk being Gell-Mann matrices, the collective coordinates matrix − 0 A(t) is written usually in the form A = ASU2 exp(iνλ4)ASU2 exp(iρλ8/√3). The Wess- Zumino term contribution into lagrangian can be calculated explicitly for this ansatz, L = ω N B/2√3, and so called ”right” hypercharge, or hypercharge in the body- W Z − 8 c fixed system equals Y = 2∂L/∂ω /√3 = N B/3. For any SU(3) multiplet (p, q) the R − 8 c maximal hypercharge Ymax = (p + 2q)/3, and obviously, inequality should be fulfilled p + 2q NcB, or ≥ p + 2q = 3(B + m) (2) for Nc = 3, with m positive integer. States with m = 0 can be called, naturally, minimal multiplets. For B = 1 they are well known octet (1, 1) and decuplet (3, 0) [28]. States with m = 1 should contain at least one qq¯ pair, since they contain the S = +1, Y = 2 hyperon. They are pentaquarks antidecuplet (p, q) = (0, 3), 27-plet (2, 2), 35-plet (4, 1). 28-plet (6, 0) should contain already 2 quark-antiquark pairs, as it follows from analysis of its isospin content, so it is septuquark (or ) [46]. The pentaquark multiplets are presented at Fig. 1. The minimal value of hypercharge is Ymin = (2p + q)/3, the maximal isospin Imax = (p + q)/2 at Y = (p q)/3. Such multiplets−as 27 , 35 for m = 1 and multiplets for m = 2 in their in−ternal points contain 2 or more{ } states{ } (shown by double or triple circles in Fig.1). The 28-plet (p = 6, q = 0) should contain at least two quark-antiquark pairs, so, it is septuquark (or heptaquark) although it has m = 1, and it is not shown in Fig. 1.

4 The mass formula

The lagrangian describing baryons or baryonic system is quadratic form in angular velocities defined above, with momenta of inertia, isotopical (pionic) Θπ and flavor, or kaonic ΘK as coefficients [28]:

1 2 2 2 1 2 2 NcB Lrot = Θπ(ω + ω + ω ) + ΘK (ω + ... + ω ) ω8. (3) 2 1 2 3 2 4 7 − 2√3 The expressions for these moments of inertia as functions of skyrmion profile are well known [48, 28] and presented in many papers, see e.g. [46]. The quantization condition (1) discussed above follows from the presence of linear in angular velocity ω8 term in (3) originated from the Wess-Zumino-Witten term in the action of the model [44, 28]. The hamiltonian of the model can be obtained from (3) by means of canonical quantization procedure [28]:

2 2 1 2 1 2 Nc B H = Mcl + R~ + C2(SU3) R~ , (4) 2Θπ 2ΘK  − − 12  8 2 where the second order Casimir operator for the SU(3) group, C2(SU3) = a=1 Ra, with eigenvalues for the (p, q) multiplets C (SU ) = (p2 + pq + q2)/3 + p + q, for the 2 3 p,q P ~ 2 2 2 2 SU(2) group, C2(SU2) = R = R1 + R2 + R3 = J(J + 1) = IR(IR + 1). The operators Rα = ∂L/∂ωα satisfy definite commutation relations which are generalization of the angular momentum commutation relations to the SU(3) case [28]. Evidently, the linear in ω terms in lagrangian (3) are cancelled in hamiltonian (4). The equality of angular momentum (spin) J and the so called right or body fixed isospin IR used in (4) takes place only for configurations of the ”hedgehog” type when usual space and isospace rotations are equivalent. This equality is absent for configurations which provide the minimum of classical energy for greater baryon numbers, B 2. ≥ For minimal multiplets (m = 0) the right isospin IR = p/2, and it is easy to check that coefficient of 1/2ΘK in (5) equals to K = C (SU ) R~ 2 N 2 B2/12 = N B/2, (5) 2 3 − − C C 1 for arbitrary NC . So, K is the same for all multiplets with m = 0 [29], see Table 1- the property known long ago for the B = 1 case [28]. For nonminimal multiplets 2 there are additional contributions to the energy proportional to m/ΘK and m /ΘK, according to (4)[29]. It means that in the framework of chiral soliton approach the ”weight” of quark- antiquark pair is defined by parameter 1/ΘK, and this property of such models deserves better understanding. (p, q) N(p, q) m C (SU ) J = I K(J ) K(J 1) 2 3 R max max − (1, 1) 8 0 3 1/2 3/2 (3, 0) {10} 0 6 3/2 3/2 { } (0, 3) 10 1 6 1/2 3/2+3 (2, 2) {27} 1 8 3/2; 1/2 3/2+2 3/2+5 { } (4, 1) 35 1 12 5/2; 3/2 3/2+1 3/2+6 (6, 0) {28} 1 18 5/2 3/2+7 { } (1, 4) 35 2 12 3/2; 1/2 3/2+6 3/2+9 (3, 3) {64} 2 15 5/2; 3/2; 1/2 3/2+4 3/2+9 (5, 2) {81} 2 20 7/2; 5/2; 3/2 3/2+2 3/2+9 { } (7, 1) 80 2 27 7/2; 5/2 3/2+9 3/2+16 (9, 0) {55} 2 36 7/2 3/2+18 { } Table 1.The values of N(p, q), Casimir operator C2(SU3), spin J = IR, coefficient K for first two values of J for minimal (m = 0) and nonminimal (m = 1, 2) multiplets of baryons.

It follows from Table 1 [46] that for each nonzero m the coefficient K(Jmax) decreases with increasing N(p, q), e.g. K5/2(35) < K3/2(27) < K1/2(10). The following differences of the rotation energy can be obtained easily: 3 3 M10 M8 = , M10¯ M8 = , (6) − 2Θπ − 2ΘK obtained in [28, 31], 1 3 1 M27,J=3/2 M10 = , M27,J=3/2 M10¯ = , (7) − ΘK − 2Θπ − 2ΘK 5 1 M35,J=5/2 M27,J=3/2 = . (8) − 2Θπ − 2ΘK

If the relation took place ΘK Θπ then 27 -plet would be lighter than antidecuplet, and 35 -plet would be lighterthan 27 {. In}realistic case Θ is approximately twice { } { } K smaller than Θπ (see Table 2, next section), and therefore the components of antide- cuplet are lighter than components of 27 with same values of strangeness. Beginning with some values of N(p, q) coefficient{K increases} strongly, as can be seen from Table 1, and this corresponds to the increase of the number of quark-antiquark pairs by another unity. The states with J < Jmax have the energy considerably greater than that of Jmax states, by this reason they could contain also greater amount of qq¯-pairs. The formula (5) is obtained in the rigid rotator approximation which is valid if the profile function of the skyrmion and therefore its dimensions and other properties are not changed when it is rotated in the configuration space (see, e.g. discussion in [46].

1 It should be kept in mind that for NC different from 3 the minimal multiplets for baryons differ from octet and decuplet. They have (p, q) = (1, (NC 1)/2), (3, (NC 3)/2), ..., (NC , 0). − − 5 Spectrum of baryonic states

Expressions (4), (5) and numbers given in Table 1 are sufficient to calculate the spectrum of baryons without mass splitting inside of SU(3)- multiplets, as it was made e.g. in [25, 29]. The mass splitting due to the presence of flavor symmetry breaking terms plays a very substantial role [30, 31, 33]:

1 D(8) H = − 88 Γ (9) SB 2 SB where the SU(3) rotation function D8 (ν) = 1 3s2/2, 88 − ν 2 2 FK 2 2 2 2 ˜ ΓSB = 2 mK mπ Σ + (FK Fπ )Σ (10) 3" Fπ − ! − #

2 2s2 Fπ 3 ˜ 1 02 f 3 Σ = (1 cf )d ~r, Σ = cf f + 2 d r, (11) 2 Z − 4 Z r  kaon and masses mK, mπ are taken from experiment. The quantity SC =< 2 (8) sν > /2 =< 1 D88 > /3 averaged over the baryon SU(3) wave function defines its strangeness −content. Without configuration mixing, i.e. when flavor symmetry 2 breaking terms in the lagrangian are considered as small perturbation, < sν >0 can be expressed simply in terms of the SU(3) Clebsh-Gordan coefficients. The values of 2 < sν >0 for the octet, decuplet, antidecuplet and some components of higher multiplets are presented in Table 2. In this approximation the components of 10 and 10 are placed equidistantly, and splittings of decuplet and antidecuplet are{equal.} { } The spectrum of states with configuration mixing and diagonalization of the hamiltonian in the next order of perturbation theory in HSB is given in Table 2 (the code for calculation was kindly presented by H.Walliser). The calculation results in the Skyrme model with only one adjustable parameter - Skyrme constant e (Fπ = 186 Mev - experimentally measured value) are shown as variants A and B. The values 2 of < sν > become lower when configuration mixing takes place, and equidistant spacing of components inside of decuplet and especially antidecuplet is violated, see also Fig.2. It should be stressed here that the chiral soliton approach in its present state can describe the differences of baryon or multibaryon masses [45, 30, 33]. The absolute 0 values of mass are controlled by loop corrections of the order of NC 1 which are estimated now for the case of B = 1 only [47]. Therefore, the value of∼nucleon mass in Table 2. and Fig.2 is taken to be equal to the observed value. As it can be seen from Table 2, the agreement with data for pure Skyrme model with one parameter is not so good, but the observed mass of Θ+ is reproduced with some reservation. To get more reliable predictions for masses of other exotic states the more phenomenological approach was used in [33] where the observed value MΘ = 1.54 Gev was included into the fit, and ΘK, ΓSB were the variated parameters (variant C in Table 2 and Fig.2). The position of some components of 27 , 35 and 28 plets is shown as well. { } { } { It}looks astonishing at first sight that the state Θ+ containing strange antiquark is lighter than nonstrange component of antidecuplet, N ∗(I = 1/2). But it is easy to understand if we recall that all antidecuplet components contain qq¯ pair: Θ+ contains 4 light quarks and s¯, N ∗ contains 3 light quarks and ss¯ pair with some weight, Σ∗ 10¯ contains u, d, s quarks and ss¯, etc. ∈ { } A B C −1 Θπ (Gev ) — 6.175 5.556 5.61 - −1 ΘK (Gev ) — 2.924 2.641 2.84 - ΓSB (Gev) — 1.369 1.244 1.45 - Baryon N, Y, I, J > < s2 > A B C Data | ν 0 Λ 8, 0, 0, 1/2 > 0.60 1094 1078 1103 1116 | Σ 8, 0, 1, 1/2 > 0.73 1202 1182 1216 1193 Ξ |8, 1, 1/2, 1/2 > 0.80 1310 1274 1332 1318 | − ∆ 10, 1, 3/2, 3/2 > 0.58 1228 1258 1253 1232 Σ∗|10, 0, 1, 3/2 > 0.67 1357 1372 1391 1385 Ξ∗ |10, 1, 1/2, 3/2 > 0.75 1483 1484 1525 1530 | − Ω 10, 2, 0, 3/2 > 0.83 1604 1587 1654 1672 | − Θ+ 10, 2, 0, 1/2 > 0.50 1520 1564 1539 1540 N ∗ |10, 1, 1/2, 1/2 > 0.58 1663 1664 1661 1710? Σ∗ |10, 0, 1, 1/2 > 0.67 1731 1749 1764 1770? ∗ | Ξ3/2 10, 1, 3/2, 1/2 > 0.75 1753 1781 1786 1862? ∗ | − Θ1 27, 2, 1, 3/2 > 0.57 1646 1697 1688 Σ∗ |27, 0, 2, 3/2 > 0.61 1675 1724 1718 2 | Ξ∗ 27, 1, 3/2, 3/2 > 0.71 1798 1835 1850 1862? 3/2| − Ω∗ 27, 2, 1, 3/2 > 0.82 1928 1950 1987 1 | − Θ∗ 35, 2, 2, 5/2 > 0.71 1979 2055 2061 2 | ∆5/2 35, 1, 5/2, 5/2 > 0.44 1723 1817 1792 ∗ | Σ2 35, 0, 2, 5/2 > 0.54 1842 1924 1918 ∗ | Ξ3/2 35, 1, 3/2, 5/2 > 0.65 1963 2032 2046 ∗ | − Ω1 35, 2, 1, 5/2 > 0.75 2085 2141 2175 |35, −3, 1/2, 5/2 > 0.85 2208 2251 2306 | − Table 2. Values of masses of the octet, decuplet, antidecuplet and manifestly exotic components of higher multiplets. A: e = 3.96; B: e = 4.12; C: fit with parameters ΘK , Θπ and ΓSB [33], which are shown in the upper 3 lines. The mass splitting inside of decuplet is influenced essentially by its mixing with 27 -plet components [33], see Fig.1, which increases this splitting considerably - the {effect} ignored in [31]. The mixing of antidecuplet with the octet of baryons has ∗ ∗ ∗ some effect on the position of N and Σ , the position of Ξ3/2 is influenced strongly by mixing with 27 -plet (J = 1/2) and 35¯ -plet. As a result of mixing, the mass splitting of antidecuplet{ } decreases. Position{ of}Θ∗ 27 is influenced by mixing with higher multiplets [33], the components of 35 -plet∈ {mix} mainly with corresponding components of septuquark 64 -plet. { } { } ∗ It should be noted that predictions of the mass of Ξ3/2 made in [33] half a year before its observation at CERN [13] were quite close to the reported value 1862 Mev: 1786 Mev for the component of antidecuplet, and 1850 Mev for the 27 component, variant C of Table 2. The component of 35 -plet with zero strangeness{ }and I = J = { } 2 5/2 is of special interest because it has the smallest strangeness content (or sν) - smaller than nucleon and ∆. It is the lightest component of 35 -plet, and this remarkable property has explanation in simplistic pentaquark mo{del,}see Section 6 below. As a consequence of isospin conservation by strong interactions it can decay into ∆π, but not to Nπ or Nρ. According to Table 1, the components of 28 plet containing 2 qq¯ pairs at least, have the mass considerably greater than that{of }other multiplets of Fig.1. All baryonic states considered here are obtained by means of quantization of soliton rotations in SU(3) configuration space, therefore they have positive parity. A qualitative discussion of the influence of other (nonzero) modes - vibration, breathing - as well as references to corresponding papers can be found in [33, 32]. The realistic situation can be more complicated than somewhat simplified picture presented here, since each rotation state can have vibrational excitations with characteristic energy of hundreds of Mev. If the matrix element of the decay Θ+ KN is written in a form → † MΘ→KN = gΘKN u¯N γ5uΘφK (12) with uN and uΘ - bispinors of final and initial baryons, then the decay width equals to

2 2 2 2 cm 3 gΘKN ∆M mK cm gΘKN (pK ) ΓΘ→KN = −2 pK (13) 8π M ' 8π MmN

cm where ∆M = M mN , M is the mass of decaying baryon, pK - the kaon momentum in the c.m. frame.− For the decay constant we obtain then g 4.4 if we take the ΘKN ' value ΓΘ→KN = 10 Mev as suggested by experimental data [2],[16]. This should be compared with gπNN 13.5. So, suppression of the decay Θ KN takes place, but not large and understandable,' according to [31, 35, 36]. It would→ be difficult, however, to explain the with Γ 1 Mev or smaller, as suggested by scattering data [11, 12]. Θ ∼ 6 Comments on wave functions of pentaquarks

Wave functions (WF) of manifestly exotic resonances are unique within pentaquark approximation, i.e. their quark content is fixed by isospin, strangeness, etc. It is easily to obtain for manifestly exotic components of antidecuplet (see also Fig. 1):

Ψ uudds¯, Θ ∼ Ψ ssddu¯, ... , ... , ssuud¯, Ξ3/2 ∼ WF of cryptoexotic states are not unique. Within antidecuplet:

Ψ ∗ udd [α uu¯ + β dd¯+ γ ss¯], uud [α uu¯ + β dd¯+ γ ss¯], N ∼ − − − + + + Ψ ∗ sdd [µ uu¯ + ν dd¯+ ρ ss¯], ... , sdd [µ uu¯ + ν dd¯+ ρ ss¯], Σ ∼ − − − + + + coefficients α−, α+, etc depend on the particular variant of the model. E.g., for the diquark model with D 3¯ [36] α = 1/3, β = 0, γ = 2/3, etc. q ∼ F − − − Equidistancy within 10 was obtained in [37]. q q Within 27 -plet only S = 0, I = 3/2-state (analog of ∆-isobar) is cryptoexotic. The states with{ S}= +1, I = 1 and state with S = 1, I = 2 contain one s-quark field as depicted in Fig. 1, and their masses do not differ− much by this reason, as it was obtained in chiral soliton model as well, see Table 2, Fig. 2. Within 35-plet ALL states of maximal isospin are manifestly exotic and have unique quark content. The state with S = 0, I = 5/2 (it can be called ∆5/2) does not contain strange quarks: Ψ ddddu¯, ... , uuuud¯, ∆5/2 ∼ neither s, nor s¯ quarks! Remarkably, that within chiral soliton model this state has minimal strangeness content and has the lowest (within 35 -plet) mass, see Table 2. Besides flavor antitriplet diquark D 3¯ discussed{ in} this context first in [36], q ∼ F the diquarks Dq 6F are necessary to form 27 - and 35 -plets of pentaquarks. The comparison of masses∼ of Θ+ 10¯ and Θ∗ {27 }allows {to conclude} that 6 diquark is ∈ ∈ F heavier that 3¯F by 120 150 Mev, at least. To conclude this section I note that there is good correspondence b−etween chiral soliton calculations [33] and simplistic pentaquark model, the fact which needs probably deeper understanding. 7 Conclusions and prospects

Even if not all reported states are confirmed, one can state that new and interesting branch of (baryon) spectroscopy appeared which will enlarge our knowledge about hadron structure. The following problems and questions can be pointed out: High statistics confirmation of existence of narrow pentaquarks seems to be ∗ necessary, especially for the resonances Ξ3/2 and Θc, see [19]. Width determination is of great importance, Γ 1 Mev is not excluded and suggested∗ by analyses of scattering data, but would be difficult∼ to explain by theory: a special reason is necessary then. Several missing components of multiplets remain to be found, for example: ¯ ∗ + 0 + + ¯ 0 in 10 -plet: Ξ3/2 Ξ π , Σ K ; { } ∗ → ∗ ¯ in 27 -plet: Θ1 NK; Σ2 Σ π; Ξ3/2; Ω1 Ω π, ΞK; { } ∗ → ¯ → → ¯ in 35 -plet: Ω1 Ω π, ΞK; ∆5/2 Nππ; ΞS=−4 ΩK, etc. { } Studies →of cryptoexotics (N→∗, ∆∗, Ξ∗...) are→ of interest as well, to complete the picture∗ of pentaquarks. Spin and parity are crucial for cheking the validity of chiral soliton models predictions.∗ Negative parity of these states would provide big difficulties for their interpretation as quantized topological solitons. As a result, better understanding of the structure of baryons wave functions will be∗reached. Other predictions of CSM are of interest, e.g. supernarrow radiatively decay- ing dibary∗ on [49] (JINR and INR experiments [50, 51], see, however, [52]); Θ+ hyper- 3 4 nuclei and anti-charmed hypernuclei, e.g. Hc¯ ddddduuuuuc¯, Hec¯, etc. [46, 53]. I’m indebted to V.A.Andrianov, B.L.Ioffe,∼ L.N.Lipatov for numerous discus- sions during the conference. References

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(dddds¯) u u u u u(uuuus¯) ¡ A ¡ A ¡ A ¯ (ddddu¯) u¡ fs fs fs fs Au(uuuud) A ¡ A ¡ A ¡(uuusd¯) (dddsu¯)Au fs fs fs ¡u - A ¡ I3 A ¡ (ddssu¯)A ¡(uussd¯) Au fs fs ¡u A ¡ A ¡ ¯ (dsssu¯)Au fs ¡u(usssd) A ¡ A ¡ A ¡ ¯ (ssssu¯)Au u¡(ssssd)

35 J = 5/2; 3/2 { }

Figure 1: The I Y diagrams for the multiplets of pentaquarks, B = 1, m = 1. 3 − Large full circles show the exotic states, smaller - the cryptoexotic states which can mix with nonexotic states from octet and decuplet. Manifestly exotic components of pentaquarks have unique quark contents shown in the figure. C D exp C D exp C D (Y I ) C D (Y I ) C D (Y I )

2 (-2 1) (-1 1/2) (2 2) (0 0) (0 2) (0 1) (-1 3/2) (-1 3/2) (1 1/2) (1 5/2) (0 1) (1 3/2) (0 2) (1 1/2) (2 1)

(2 0) 1.5 ENERGY [GeV]

1

{8} J=1/2 {10} J=3/2 {10} J=1/2 {27} J=3/2 {35} J=5/2

Figure 2: Lowest rotational states in the SU(3) soliton model for fits C and D. The experimental masses of the 8 and 10 baryons are depicted for comparison. Not all { } { } states of the 35 are shown. This figure is taken from [33]. The variant D shown here takes into accoun{ }t the term in H which appears from the ρ ω mixing in effective SB − lagrangian [30, 33].